Question

The angle between the line `bar"r" = (hat"i" + 2hat"j" + hat"k") + lambda(hat"i" + hat"j" + hat"k")` and the plane `bar"r"*(2hat"i" - hat"j" + hat"k")` = 8 is ______.

Options

• `pi/2`

• `sin^-1(sqrt(2)/3)`

• `pi/4`

• `sin^-1(1/3)`

Answer 1

The angle between the line `bar"r" = (hat"i" + 2hat"j" + hat"k") + lambda(hat"i" + hat"j" + hat"k")` and the plane `bar"r"*(2hat"i" - hat"j" + hat"k")` = 8 is `sin^-1(sqrt(2)/3)`.

Explanation:

`bar"b" = hat"i" + hat"j" + hat"k"` and `bar"n" = 2hat"i" - hat"j" + hat"k"`

sin θ = `|(bar"b" * bar"n")/(|bar"b"|*|bar"n"|)|`

= `|((1)(2) + (1)(-1) + (1)(1))/(sqrt(1^2 + 1^2 + 1^2) sqrt(2^2 + (-1)^2 + 1^2))|`

∴ sin θ = `|(2 - 1 + 1)/(sqrt(3) sqrt(6))| = sqrt(2)/3`

⇒ θ = `sin^-1(sqrt(2)/3)`